The complexity of a numerical semigroup

Abstract

Let $S$ and $\Delta$ be numerical semigroups. A numerical semigroup $S$ is an $\mathbf{I}(\Delta )$-{\it semigroup} if $S\backslash \{0\}$ is an ideal of $\Delta$. We will denote by $\mathcal{J}(Δ)=\{S \mid S \text{ is an $\mathbf{I}(Δ)$-semigroup} \}.$ We will say that $\Delta$ is {\it an ideal extension of } $S$ if $S\in \mathcal{J}(\Delta ).$ In this work, we present an algorithm that allows to build all the ideal extensions of a numerical semigroup. We can recursively denote by ${\mathcal{J}}^0(\mathbb{N})=\mathbb{N},$ ${\mathcal{J}}^1(\mathbb{N})=\mathcal{J}(\mathbb{N})$ and ${\mathcal{J}}^{k+1}(\mathbb{N})=\mathcal{J}({\mathcal{J}}^k(\mathbb{N}))$ for all $k\in \mathbb{N}.$ The complexity of a numerical semigroup $S$ is the minimun of the set $\{k\in \mathbb{N}\mid S\in {\mathcal{J}}^k(\mathbb{N})\}.$ In addition, we will give an algorithm that allows us to compute all the numerical semigroups with fixed multiplicity and complexity.